Celebratio Mathematica

A. Adrian Albert

Abraham Adrian Albert: 1905–1972

Ab­ra­ham Ad­ri­an Al­bert was an out­stand­ing fig­ure in the world of
twen­ti­eth-cen­tury al­gebra, and at the same time a states­man and lead­er in the Amer­ic­an
math­em­at­ic­al com­munity. He was born in Chica­go on Novem­ber 9, 1905, the son of im­mig­rant
par­ents. His fath­er, Eli­as Al­bert, had come to the United States from Eng­land and had es­tab­lished
him­self as a re­tail mer­chant. His moth­er, Fan­nie Fradkin Al­bert, had come from Rus­sia.
Ad­ri­an Al­bert was the second of three chil­dren, the oth­ers be­ing a boy and a girl;
in ad­di­tion, he had a half-broth­er and a half-sis­ter on his moth­er’s side.

Al­bert at­ten­ded ele­ment­ary schools in Chica­go from 1911 to 1914. From 1914
to 1916 the fam­ily lived in Iron Moun­tain, Michigan, where he con­tin­ued his school­ing.
Back in Chica­go, he at­ten­ded Theodore Herzl Ele­ment­ary School, gradu­at­ing in 1919, and the John Mar­shall High School, gradu­at­ing in 1922. In the fall of 1922 he entered the Uni­versity
of Chica­go, the in­sti­tu­tion with which he was to be as­so­ci­ated for vir­tu­ally
the rest of his life. He was awar­ded the Bach­el­or of Sci­ence, Mas­ter of Sci­ence,
and Doc­tor of Philo­sophy in three suc­cess­ive years: 1926, 1927, and 1928.

On Decem­ber 18, 1927, while com­plet­ing his dis­ser­ta­tion, he mar­ried Frieda Dav­is. Theirs was a happy mar­riage, and
she was a stal­wart help to him throughout his ca­reer. She re­mains
act­ive in the Uni­versity of Chica­go com­munity and in the life of its De­part­ment
of Math­em­at­ics. They had three chil­dren: Alan, Roy, and Nancy. Tra­gic­ally, Roy died
in 1958 at the early age of twenty-three. There are five grand-chil­dren.

Le­onard Eu­gene Dick­son
was at the time the dom­in­ant Amer­ic­an math­em­atician in the fields
of al­gebra and num­ber the­ory. He had been on the Chica­go fac­ulty since al­most
its earli­est days. He was a re­mark­ably en­er­get­ic and force­ful man (as I can per­son­ally
testi­fy, hav­ing been a stu­dent in his num­ber the­ory course years later). His in­flu­ence on Al­bert was con­sid­er­able and set the course for much of his sub­sequent re­search.

Dick­son’s im­port­ant book, Al­geb­ras and Their Arith­met­ics[e1]
(Chica­go: Univ. of Chica­go Press, 1923), had re­cently ap­peared in an ex­pan­ded Ger­man trans­la­tion
[e2]
(Zurich: Orell Füss­li, 1927). The sub­ject of al­geb­ras had ad­vanced to the cen­ter of the stage. It con­tin­ues to this day to play a vi­tal role in many branches of math­em­at­ics and in oth­er sci­ences as well.

An al­gebra is an ab­stract math­em­at­ic­al en­tity with ele­ments and op­er­a­tions ful­filling the fa­mil­i­ar laws of al­gebra, with one im­port­ant qual­i­fic­a­tion — the com­mut­at­ive law of mul­ti­plic­a­tion is waived. (More care­fully, I should have said that this is an as­so­ci­at­ive al­gebra; non-as­so­ci­at­ive al­geb­ras will play an im­port­ant role later in this mem­oir.) Early in the twen­ti­eth cen­tury, fun­da­ment­al res­ults of
J. H. M. Wed­der­burn
had cla­ri­fied the nature of al­geb­ras up to the clas­si­fic­a­tion of the ul­ti­mate build­ing blocks, the di­vi­sion al­geb­ras.
Ad­vances were now needed on two fronts. One wanted
the­or­ems val­id over any field (every al­gebra has an un­der­ly­ing field of
coef­fi­cient — a num­ber sys­tem
of which the lead­ing ex­amples are the real num­bers, the ra­tion­al num­bers, and the
in­tegers mod \( p \)). On the oth­er front, one sought to clas­si­fy di­vi­sion al­geb­ras over
the field of ra­tion­al num­bers.

Al­bert at once be­came ex­traordin­ar­ily act­ive on both bat­tle­fields. His first
ma­jor pub­lic­a­tion was an im­prove­ment of the second half of his Ph.D. thes­is
[1];
it ap­peared in 1929 un­der the title “A de­term­in­a­tion of all nor­mal di­vi­sion al­geb­ras in six­teen units”
[2].
The hall­marks of his math­em­at­ic­al per­son­al­ity were already vis­ible. Here was a tough
prob­lem that had de­feated his pre­de­cessors; he at­tacked it with tenacity till it
yiel­ded. One can ima­gine how de­lighted Dick­son must have been. This work won Al­bert a pres­ti­gi­ous postdoc­tor­al Na­tion­al Re­search Coun­cil Fel­low­ship, which he used in
1928 and 1929 at Prin­ceton and Chica­go.

I shall briefly ex­plain the nature of Al­bert’s ac­com­plish­ment. The di­men­sion of a di­vi­sion al­gebra over its cen­ter
is ne­ces­sar­ily a square, say \( n^2 \). The case \( n = 2 \) is easy. A good deal harder is the case
\( n = 3 \), handled by Wed­der­burn. Now Al­bert cracked the still harder case, \( n = 4 \). One
in­dic­a­tion of the mag­nitude of the res­ult is the fact that at this writ­ing, nearly
fifty years later, the next case \( (n = 5) \) re­mains mys­ter­i­ous.

In the hunt for ra­tion­al di­vi­sion al­geb­ras, Al­bert had stiff com­pet­i­tion. Three top Ger­man al­geb­ra­ists
(Richard Brauer,
Helmut Hasse,
and
Emmy No­eth­er)
were after the same big game (just a little later the ad­vent of the Nazis brought
two-thirds of this stel­lar team to the United States.) It was an un­equal battle, and
Al­bert was nosed out in a photo fin­ish.
In a joint pa­per
[◊]
with Hasse
pub­lished in 1932
the full his­tory of the mat­ter was set out, and one can see how close Al­bert came to win­ning.

Let me re­turn to 1928–1929, his first postdoc­tor­al year. At Prin­ceton Uni­versity a for­tu­nate con­tact took place.
So­lomon Lef­schetz
noted the pres­ence of this prom­ising young­ster, and en­cour­aged him to take a look at Riemann matrices. These are matrices that arise in the the­ory of com­plex man­i­folds; the main prob­lems con­cern­ing them had re­mained un­solved for more than half a cen­tury. The pro­ject was per­fect for Al­bert, for it con­nec­ted closely with the the­ory of al­geb­ras he was so suc­cess­fully de­vel­op­ing. A series of pa­pers en­sued, cul­min­at­ing in com­plete solu­tions of the out­stand­ing prob­lems con­cern­ing Riemann matrices. For this work he re­ceived the Amer­ic­an Math­em­at­ic­al So­ci­ety’s 1939 Cole prize in al­gebra.

From 1929 to 1931 he was an in­struct­or at Columbia Uni­versity. Then, the young
couple, ac­com­pan­ied by a baby boy less than a year old, hap­pily re­turned to the Uni­versity
of Chica­go. He rose stead­ily through the ranks: as­sist­ant pro­fess­or in 1931, as­so­ci­ate pro­fess­or in 1937,
pro­fess­or in 1941, chair­man of the De­part­ment of Math­em­at­ics from 1958 to 1962, and
dean of the Di­vi­sion of Phys­ic­al Sci­ences from 1962 to 1971. In 1960 he re­ceived a Dis­tin­guished
Ser­vice Pro­fess­or­ship, the highest hon­or that the Uni­versity of Chica­go can con­fer
on a fac­ulty mem­ber; ap­pro­pri­ately, it bore the name of
E. H. Moore,
chair­man of the De­part­ment from its first day un­til 1927.

The dec­ade of the 1930s saw a cre­at­ive out­burst. Ap­prox­im­ately sixty pa­pers flowed
from his pen. They covered a wide range of top­ics in al­gebra and the the­ory of num­bers bey­ond
those I have men­tioned. Some­how, he also found the time to write two im­port­ant books.
Mod­ern High­er Al­gebra[5]
(1937)
was a widely used
text­book — but
it is more than a text­book. It re­mains in
print to this day, and on cer­tain sub­jects it is an in­dis­pens­able ref­er­ence.
Struc­ture of Al­geb­ras[6]
(1939)
was his defin­it­ive treat­ise on al­geb­ras and
formed the basis for his 1939 Col­loqui­um Lec­tures to the Amer­ic­an Math­em­at­ic­al
So­ci­ety. There have been later books on al­geb­ras, but none has re­placed
Struc­ture of Al­geb­ras.

The aca­dem­ic year 1933–1934 was again spent in Prin­ceton, this
time at the newly foun­ded In­sti­tute for Ad­vanced Study. Again, there were fruit­ful
con­tacts with oth­er math­em­aticians. Al­bert has re­cor­ded that he found
Her­mann Weyl’s
lec­tures on Lie al­geb­ras stim­u­lat­ing. An­oth­er thing that happened was that Al­bert was in­tro­duced to Jordan al­geb­ras.

The phys­i­cist
Pas­cu­al Jordan
had sug­ges­ted that a cer­tain kind of al­gebra, in­spired
by us­ing the op­er­a­tion \( xy + yx \) in an as­so­ci­at­ive al­gebra, might be use­ful in quantum
mech­an­ics. He en­lis­ted
von Neu­mann
and
Wign­er
in the en­ter­prise, and in a joint pa­per they in­vest­ig­ated the struc­ture in ques­tion. But a cru­cial point was left un­re­solved; Al­bert sup­plied the miss­ing the­or­em. The pa­per ap­peared in 1934 and was en­titled “On a cer­tain al­gebra of quantum mech­an­ics”
[4].
A seed had been planted that Al­bert was to har­vest a dec­ade later.

Let me jump ahead chro­no­lo­gic­ally to fin­ish the story of Jordan al­geb­ras.
I can add a per­son­al re­col­lec­tion. I ar­rived in Chica­go in early Oc­to­ber 1945. Per­haps
on my very first day, per­haps a few days later, I was in Al­bert’s of­fice dis­cuss­ing some routine mat­ter. His stu­dent
Daniel Zel­in­sky
entered. A tor­rent of words poured out, as Al­bert told him how he had just cracked the the­ory of spe­cial Jordan al­geb­ras. His en­thu­si­asm was de­light­ful and con­ta­gious. I got in­to the act and we had a spir­ited dis­cus­sion. It res­ul­ted in arous­ing in me
an en­dur­ing in­terest in Jordan al­geb­ras.

About a year later, in 1946, his pa­per ap­peared. It was fol­lowed by
“A struc­ture the­ory for Jordan al­geb­ras”
[9]
(1947)
and
“A the­ory of power-as­so­ci­at­ive com­mut­at­ive al­geb­ras”
[11]
(1950).
These three pa­pers cre­ated a whole sub­ject; it was an achieve­ment com­par­able to his study of Riemann matrices.

World War II brought changes to the Chica­go cam­pus. The Man­hat­tan Pro­ject took over Eck­hart
Hall, the math­em­at­ics build­ing (the self-sus­tain­ing chain re­ac­tion of
Decem­ber 1942 took place a block away). Sci­ent­ists in all dis­cip­lines, in­clud­ing
math­em­at­ics, answered the call to aid the war ef­fort against the Ax­is. A num­ber
of math­em­aticians as­sembled in an Ap­plied Math­em­at­ics Group at
North­west­ern Uni­versity, where Al­bert served as as­so­ci­ate dir­ect­or dur­ing 1944 and
1945. At that time, I was a mem­ber of a sim­il­ar group at Columbia, and our first sci­entif­ic
in­ter­change took place. It con­cerned a math­em­at­ic­al ques­tion arising in aer­i­al pho­to­graphy;
he gently guided me over the pit­falls I was en­coun­ter­ing.

Al­bert be­came in­ter­ested in cryp­to­graphy. On Novem­ber 22, 1941, he gave an
in­vited ad­dress at a meet­ing of the Amer­ic­an Math­em­at­ic­al So­ci­ety in Man­hat­tan, Kan­sas, en­titled
“Some math­em­at­ic­al as­pects of cryp­to­graphy.”1
After the war he con­tin­ued to
be act­ive in the fields in which he had be­come an ex­pert.

In 1942 he pub­lished a pa­per en­titled “Non-as­so­ci­at­ive al­geb­ras”
[8],
[7].
The date of re­ceipt was Janu­ary 5, 1942, but he had already presen­ted it to the Amer­ic­an Math­em­at­ic­al So­ci­ety on Septem­ber 5, 1941, and he had lec­tured on the sub­ject at Prin­ceton and Har­vard
dur­ing March of 1941. It seems fair to name one of these present­a­tions the birth
date of the Amer­ic­an school of non-as­so­ci­at­ive al­geb­ras, which he single­han­dedly foun­ded.
He was act­ive in it him­self for a quarter of a cen­tury and the school con­tin­ues to flour­ish.

Al­bert in­vest­ig­ated just about every as­pect of non-as­so­ci­at­ive al­geb­ras. At times a par­tic­u­lar line of at­tack failed
to ful­fill the prom­ise it had shown; he would then ex­er­cise his sound in­stinct and
good judg­ment by shift­ing the as­sault to a dif­fer­ent area. In fact, he re­peatedly
dis­played an un­canny knack for se­lect­ing pro­jects which later turned out to be well
con­ceived as the fol­low­ing three cases il­lus­trate.

In the
1942
pa­per
[8]
he in­tro­duced the new concept of iso­topy. Much later it was found to be ex­actly what was needed in study­ing col­lin­eations of pro­ject­ive planes.

In a se­quence of pa­pers that began in 1952 with “On non-as­so­ci­at­ive di­vi­sion al­geb­ras,”
[13]
he in­ven­ted and stud­ied twis­ted fields. At the time, one might have thought that this was merely an ad­di­tion to the list of known non-as­so­ci­at­ive di­vi­sion al­geb­ras, a list that was already large. Just a few days be­fore this para­graph was writ­ten,
Giam­paolo Menichetti
pub­lished a proof that every three-di­men­sion­al di­vi­sion al­gebra over a fi­nite field is either as­so­ci­at­ive or a twis­ted field, show­ing con­clus­ively that Al­bert had hit on a key concept.

In a pa­per that ap­peared in 1953,
Er­win Klein­feld
clas­si­fied all simple al­tern­at­ive rings. Vi­tal use was made of two of Al­bert’s pa­pers: “Ab­so­lute-val­ued al­geb­ra­ic al­geb­ras”
[10]
(1949)
and “On simple al­tern­at­ive rings”
[12]
(1952).
I re­mem­ber hear­ing Klein­feld ex­claim “It’s amaz­ing! He proved ex­actly the right things.”

The post­war years were busy ones for the Al­berts. Just the job to be done at the Uni­versity would have ab­sorbed all the en­er­gies of a less­er man.
Mar­shall Har­vey Stone
was lured from Har­vard in 1946 to as­sume the chair­man­ship of the Math­em­at­ics De­part­ment. Soon Eck­hart Hall was hum­ming, as such world-fam­ous math­em­aticians as
Shi­ing-Shen Chern,
Saun­ders Mac Lane,
An­dré Weil,
and
Ant­oni Zyg­mund
joined Al­bert and Stone to make Chica­go an ex­cit­ing cen­ter. Al­bert taught courses at all levels, dir­ec­ted his stream of Ph.D.s, main­tained his own pro­gram of re­search, and helped to guide the De­part­ment and the Uni­versity at large in mak­ing wise de­cisions. Even­tu­ally, in 1958, he ac­cep­ted the chal­lenge of the Chair­man­ship. The main stamp he left on the De­part­ment was a pro­ject dear to his heart: main­tain­ing a lively flow of vis­it­ors and re­search in­struct­ors, for whom he skill­fully got sup­port
in the form of re­search grants. The Uni­versity co­oper­ated by mak­ing an apart­ment
build­ing avail­able to house the vis­it­ors. Af­fec­tion­ately called “the
com­pound,” the mod­est build­ing has been the birth­place of many a fine the­or­em. Es­pe­cially mem­or­able was the aca­dem­ic year 1960–1961
when
Wal­ter Feit
and
John Thompson,
vis­it­ing for the en­tire year, made their big break­through in fi­nite group the­ory by prov­ing that all groups of odd or­der are solv­able.

Early in his second three-year term as chair­man, Al­bert was asked to as­sume
the de­mand­ing post of Dean of the Di­vi­sion of Phys­ic­al Sci­ences. He ac­cep­ted, and served for nine years. The new dean was able to keep his math­em­at­ics go­ing. In 1965 he re­turned to his first love: as­so­ci­at­ive di­vi­sion al­geb­ras. His re­tir­ing pres­id­en­tial ad­dress to the Amer­ic­an Math­em­at­ic­al So­ci­ety, “On as­so­ci­at­ive di­vi­sion al­gebra”
[14]
presen­ted the state of the art as of 1968.

Re­quests for his ser­vices from out­side the Uni­versity were wide­spread and
fre­quent. A full tab­u­la­tion would be long in­deed. Here is a par­tial list: con­sult­ant,
Rand Cor­por­a­tion; con­sult­ant, Na­tion­al Se­cur­ity Agency; trust­ee, In­sti­tute for Ad­vanced Study;
trust­ee, In­sti­tute for De­fense Ana­lyses, 1969–1972, and dir­ect­or of its Com­mu­nic­a­tions
Re­search Di­vi­sion, 1961–1962; chair­man, Di­vi­sion of Math­em­at­ics of the Na­tion­al
Re­search Coun­cil, 1952–1955; chair­man, Math­em­at­ics Sec­tion of the Na­tion­al Academy of
Sci­ences, 1958–1961; chair­man, Sur­vey of Train­ing and Re­search Po­ten­tial in the Math­em­at­ic­al
Sci­ences, 1955–1957 (widely known as the “Al­bert Sur­vey”); pres­id­ent, Amer­ic­an Math­em­at­ic­al So­ci­ety, 1965–1966; par­ti­cipant and then dir­ect­or of Pro­ject SCAMP at the Uni­versity of
Cali­for­nia at Los Angeles; dir­ect­or, Pro­ject ALP (nick­named “Ad­ri­an’s little pro­ject”);
dir­ect­or, Sum­mer 1957 Math­em­at­ic­al Con­fer­ence at Bowdoin Col­lege, a pro­ject of the Air Force Cam­bridge Re­search
Cen­ter; vice-pres­id­ent, In­ter­na­tion­al Math­em­at­ic­al Uni­on; and del­eg­ate IMU Mo­scow Sym­posi­um, 1971 hon­or­ing
Vino­gradov’s eighti­eth birth­day (this was the last ma­jor meet­ing he at­ten­ded).

Al­bert’s elec­tion to the Na­tion­al Academy of Sci­ences came in 1943, when
he was thirty-sev­en. Oth­er hon­ors fol­lowed. Hon­or­ary de­grees were awar­ded by Notre Dame in
1965, by Ye­shiva Uni­versity in 1968, and by the Uni­versity of Illinois Chica­go Circle Cam­pus
in 1971. He was elec­ted to mem­ber­ship in the Brazili­an Academy of Sci­ences (1952) and
the Ar­gen­tine Academy of Sci­ences (1963).

In the fall of 1971, he was wel­comed back to the third floor of Eck­hart Hall
(the dean’s of­fice was on the first floor). He re­sumed the role of a fac­ulty mem­ber
with a zest that sug­ges­ted that it was 1931 all over again. But as the aca­dem­ic year 1971–1972 wore on,
his col­leagues and friends were saddened to see that his health was fail­ing.
Death came on June 6, 1972. A pa­per pub­lished posthum­ously in 1972 was a fit­ting coda to a life un­selfishly
de­voted to the wel­fare of math­em­at­ics and math­em­aticians.

In 1976 the De­part­ment of Math­em­at­ics in­aug­ur­ated an an­nu­al event en­titled
the Ad­ri­an Al­bert Me­mori­al Lec­tures. The first lec­turer was his long-time col­league
Pro­fess­or
Nath­an Jac­ob­son
of Yale Uni­versity.

Mrs. Frieda Al­bert was gen­er­ous in her ad­vice con­cern­ing the pre­par­a­tion
of this mem­oir. I was also for­tu­nate to have avail­able three pre­vi­ous bio­graph­ic­al
ac­counts. “Ab­ra­ham Ad­ri­an Al­bert (1905–1972),”
[e4]
by
Nath­an Jac­ob­son
(Bull. Am. Math. Soc., 80: 1075–1100),
presen­ted a de­tailed tech­nic­al ap­prais­al of Al­bert’s math­em­at­ics, in ad­di­tion to a bio­graphy and a com­pre­hens­ive bib­li­o­graphy. I also wish to thank
Daniel Zel­in­sky,
au­thor of “A. A. Al­bert”
[e3]
(Am. Math. Mon., 80:661–65),
and the con­trib­ut­ors to volume 29 of Scripta Math­em­at­ica, ori­gin­ally planned as a col­lec­tion of pa­pers hon­or­ing Ad­ri­an Al­bert on his sixty-fifth birth­day. By the time it ap­peared in 1973, the ed­it­ors had the sad task of chan­ging it in­to a me­mori­al volume; the three-page bio­graph­ic­al sketch was
writ­ten by
I. N. Her­stein.

Published as a Biographical Memoir of the National Academy of Sciences, 1980.
Also published in
A cen­tury of math­em­at­ics in Amer­ica,
partI,
edi­ted by P. Duren, R. A. As­key, and U. C. Merzbach,
His­tory of Math­em­at­ics1,
Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI),
1988.
MR1003156.

All rights reserved. Republished by permission.

Footnotes

1:The twenty-nine-page manuscript of this talk was not pub­lished,
but Chica­go’s De­part­ment of Math­em­at­ics has pre­served a copy.

The chief out­stand­ing prob­lem in the the­ory of lin­ear as­so­ci­at­ive al­geb­ras over an in­fin­ite field \( F \) is the de­term­in­a­tion of all di­vi­sion al­geb­ras. This prob­lem is equi­val­ent to that of the de­term­in­a­tion of all nor­mal di­vi­sion al­geb­ras, or al­geb­ras \( A \) in which the only ele­ments of \( A \) com­mut­at­ive with every ele­ment of \( A \) are the quant­it­ies of its ref­er­ence field \( F \).

The or­der of a nor­mal di­vi­sion al­gebra is the square of an in­teger. All nor­mal di­vi­sion al­geb­ras in \( 1^2 \), \( 2^2 \) [Dick­son 1927, p. 46], and \( 3^2 \) [Wed­der­burn 1921, p. 132] units have been de­term­ined. In this pa­per all nor­mal di­vi­sion al­geb­ras in \( 4^2 \) units, the next case, are de­term­ined and shown to be the al­geb­ras of Cecioni [1923].

The prin­cip­al prob­lem in the the­ory of lin­ear al­geb­ras is that of the de­term­in­a­tion of all nor­mal di­vi­sion al­geb­ras (of or­der \( n^2 \), de­gree \( n \)) over a field \( F \). The most im­port­ant spe­cial case of this prob­lem is the case where \( F \) is an al­geb­ra­ic num­ber field of fi­nite de­gree. It is already known that for \( n = 2 \) [Dick­son 1927, p. 45], \( n = 3 \) [Wed­der­burn 1921], \( n = 4 \) [Al­bert 1932] all such al­geb­ras are cyc­lic. We shall prove here a prin­cip­al the­or­em on al­geb­ras over al­geb­ra­ic num­ber fields:

Every nor­mal di­vi­sion al­gebra over an al­geb­ra­ic num­ber field of fi­nite de­gree is a cyc­lic (Dick­son) al­gebra.

P. Jordan, J. von Neu­mann, and E. Wign­er [1934] have dis­cussed cer­tain lin­ear real non-as­so­ci­at­ive al­geb­ras of im­port­ance in quantum mech­an­ics. Their al­geb­ras \( \mathfrak{M} \) sat­is­fy the or­din­ary pos­tu­lates for ad­di­tion, the com­mut­at­ive law for mul­ti­plic­a­tion, and the dis­tributive law, but they are non-as­so­ci­at­ive.

In the pa­per quoted above it is shown that, with a single ex­cep­tion, every al­gebra sat­is­fy­ing the above pos­tu­lates is equi­val­ent to an al­gebra \( \mathfrak{M} \) whose ele­ments are or­din­ary real matrices \( x, y,\dots \) with products \( xy \) in \( \mathfrak{M} \) defined by quasi-mul­ti­plic­a­tion,
\[ xy = \tfrac{1}{2}(x\cdot y + y\cdot x), \]
where \( x\cdot y \) is the or­din­ary mat­rix product. This single ex­cep­tion is the al­gebra \( \mathfrak{M}_3^8 \) of all three rowed Her­mitian matrices with ele­ments in the real non-as­so­ci­at­ive al­gebra \( C \) of Cay­ley num­bers.

The al­geb­ras ob­tained by quasi-mul­ti­plic­a­tion of real matrices were con­sidered in earli­er pa­pers so that, as stated by the above au­thors, this seem­ingly ex­cep­tion­al case, if proved really ex­cep­tion­al, is the only al­gebra of the above type which could lead to any new form of quantum mech­an­ics.

In the present pa­per I shall prove that \( \mathfrak{M}_3^8 \) is a new al­gebra and that it is not equi­val­ent to any al­gebra ob­tained by quasi-mul­ti­plic­a­tion of real matrices. Moreover, I shall show that the re­la­tion
\[ x(yx^2) = (xy)x^2 \]
is sat­is­fied by \( \mathfrak{M}_3^8 \).

In the second part of our study of non-as­so­ci­at­ive al­geb­ras we shall give an it­er­atie con­struc­tion of new simple al­geb­ras with a unity quant­ity. All pre­vi­ous con­struc­tions of this type have used groups of auto­morph­isms or anti-auto­morph­isms and the great gen­er­al­ity of our defin­i­tion will lie pre­cisely in that we shall be able to use in­stead al­most ar­bit­rary mul­ti­plic­at­ive groups of non-sin­gu­lar lin­ear trans­form­a­tion.

An al­gebra\( \mathfrak{A} \) over a field \( \mathfrak{F} \) is a vec­tor space over \( \mathfrak{F} \) which is closed with re­spect to a product \( xy \) which is lin­ear in both \( x \) and \( y \). The product is not ne­ces­sar­ily as­so­ci­at­ive. Every ele­ment \( x \) of \( \mathfrak{A} \) gen­er­ates a sub­al­gebra \( \mathfrak{F}[x] \) of \( \mathfrak{A} \) and we call \( \mathfrak{A} \) an al­geb­ra­ic al­gebra if every \( \mathfrak{F}[x] \) is a fi­nite-di­men­sion­al vec­tor space over \( \mathfrak{F} \).

We have shown else­where [1947] that every ab­so­lute-val­ued real fi­nite-di­men­sion­al al­gebra has di­men­sion 1, 2, 4, or 8 and is either the field \( \mathfrak{R} \) of all real num­bers, the com­plex field \( \mathfrak{C} \), the real qua­ternion al­gebra \( \mathfrak{Q} \), the real Cay­ley al­gebra \( \mathfrak{D} \), or cer­tain iso­topes without unity quant­it­ies of \( \mathfrak{Q} \) and \( \mathfrak{D} \). In the present pa­per we shall ex­tend these res­ults to al­geb­ra­ic al­geb­ras over \( \mathfrak{R} \) show­ing that every al­geb­ra­ic al­gebra over \( \mathfrak{R} \) with a unity quant­ity is fi­nite-di­men­sion­al and so is one of the al­geb­ras lis­ted above. The res­ults are ex­ten­ded im­me­di­ately to ab­so­lute-val­ued al­geb­ra­ic di­vi­sion al­geb­ras, that is, to al­geb­ras without unity quant­it­ies whose nonzero quant­it­ies form a quasig­roup.

In any study of a class of lin­ear al­geb­ras the main goal is usu­ally that of de­term­in­ing the simple al­geb­ras. The au­thor has re­cently made a num­ber of such stud­ies for classes of power-as­so­ci­at­ive al­geb­ras defined by iden­tit­ies [1948] or by the ex­ist­ence of a trace func­tion [1949], and the res­ults have been some­what sur­pris­ing in that the com­mut­at­ive simple al­geb­ras have all been Jordan al­geb­ras.

In the present pa­per we shall de­rive the reas­on for this fact. Moreover we shall de­rive a struc­ture the­ory which in­cludes the struc­ture the­ory for Jordan al­geb­ras of char­ac­ter­ist­ic \( p \).

One of our main res­ults is a gen­er­al­iz­a­tion of the Wed­der­burn–Artin The­or­em on fi­nite di­vi­sion al­geb­ras. We shall show that every fi­nite power-as­so­ci­at­ive di­vi­sion al­gebra of char­ac­ter­ist­ic\( p > 5 \)is a fi­nite field.

The re­mainder of the pa­per is de­voted to show­ing that the Wed­der­burn The­or­em for fi­nite di­vi­sion al­geb­ras de­pends upon some as­sump­tion such as power-as­so­ci­ativ­ity.

We shall study the struc­ture of a cent­ral di­vi­sion al­gebra \( \mathfrak{D} \), of odd prime de­gree \( p \) over any field \( \mathfrak{F} \) of char­ac­ter­ist­ic \( p \), which has the prop­erty that there ex­ists a quad­rat­ic ex­ten­sion field \( \mathfrak{K} \) of \( \mathfrak{F} \) such that the al­gebra\( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \)is cyc­lic over\( \mathfrak{K} \). We shall ob­tain a sim­pli­fied ver­sion of the con­di­tion that a cyc­lic al­gebra \( \mathfrak{D}_0 \), of de­gree \( p \) over \( \mathfrak{K} \), shall pos­sess the fac­tor­iz­a­tion prop­erty \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \). We shall also de­rive a new suf­fi­cient con­di­tion that such a \( \mathfrak{D} \) shall be cyc­lic over \( \mathfrak{F} \), and shall present a large class of our al­geb­ras \( \mathfrak{D}_0 \) which sat­is­fy this con­di­tion.

The Bib­li­o­graph­ic Data, be­ing a mat­ter of fact and
not cre­at­ive ex­pres­sion, is not sub­ject to copy­right.
To the ex­tent pos­sible un­der law,
Math­em­at­ic­al Sci­ences Pub­lish­ers
has waived all copy­right and re­lated or neigh­bor­ing rights to the
Bib­li­o­graph­ies on Cel­eb­ra­tio Math­em­at­ica,
in their par­tic­u­lar ex­pres­sion as text, HTML, Bib­TeX data or oth­er­wise.

The Ab­stracts of the bib­li­o­graph­ic items may be copy­righted ma­ter­i­al whose use has not
been spe­cific­ally au­thor­ized by the copy­right own­er.
We be­lieve that this not-for-profit, edu­ca­tion­al use con­sti­tutes a fair use of the
copy­righted ma­ter­i­al,
as provided for in Sec­tion 107 of the U.S. Copy­right Law. If you wish to use this copy­righted ma­ter­i­al for
pur­poses that go bey­ond fair use, you must ob­tain per­mis­sion from the copy­right own­er.